We start our inverse design optimization from a state-of-the-art topological PCS waveguide design recently introduced by He et al. [1]}. As shown in Fig. REF , this lattice structure is formed from two photonic crystals (PCs), which can be described as a triangular lattice of unit cells containing two air holes, with lattice constant \(a\)  (see Fig. REF ), or alternatively as a triangular lattice of 6 hole honeycomb unit cells with lattice constant \(\sqrt{3}a\)  and cluster radius fixed at \(R=a/\sqrt{3}\) . In each two-hole unit cell, the two circular air holes have different radii, \(r_1\)  and \(r_2\) . Above the interface, the relative positions of the smaller and bigger holes are flipped about the \(x\)  axis. Both PCs therefore have the same band structure, which contains a photonic bandgap (PBG). We define the PBG as a frequency range below the light line, where there are
no radiation modes and intrinsic losses.
In Fig. REF ,
we show the PBG properties for various
system paramaters; the PBG is shaded dark blue in panels REF (b-d).
The interface is constructed by truncating each PCS at a value of \(y\)  at which a row of holes with radius \(r_2\)  exists and positioning the PCs such that the triangular lattice is continuous across the interface.
To be consistent with He et al. [1]}, the PCS and the waveguide are modeled with a silica substrate with \(\epsilon = 2.1\) , and the slab dielectric constant is \(\epsilon = 12.0\) .
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